We consider the problem of reconstructing a measurable function over G from a countable subset ofsamples, taken accordingly to a Poisson random point process, when G is a locally compact abelian group. Thisresults are applied to the problem of approximating the inverse Radon Transform of a function.Fil: Porten, Erika. Universidad Nacional de San Martín; ArgentinaFil: Medina, Juan Miguel. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Morvidone, Marcela. Universidad Nacional de San Martín; ArgentinaVIII Congreso de Matemática Aplicada, Computacional e IndustrialLa PlataArgentinaAsociación Argentina de Matemática Aplicada Computacional e IndustrialUniversidad Nacional de Rio Cuarto. Facultad de Ciencias Exactas, Físico-Química y Naturale

Medina, Juan Miguel

Morvidone, Marcela

Porten, Erika

English

CONICET Digital

RANDOM SAMPLING OVER LCA GROUPS AND INVERSION OFTHE RADON TRANSFORMPorten Erika�, Medina Juan Miguel† and Morvidone Marcela��Centro de Matemática Aplicada, Universidad Nacional de San Martı́n, 25 de Mayo y Francia, San Martı́n, BuenosAires, Argentina, erikaporten@gmail.com, mmorvidone@unsam.edu.ar†Universidad de Buenos Aires, Facultad de Ingenierı́a, Depto. de Matemática and IAM CONICET. Paseo Colón 850,CABA, Argentina, jmedina@fi.uba.arAbstract: We consider the problem of reconstructing a measurable function over G from a countable subset ofsamples, taken accordingly to a Poisson random point process, when G is a locally compact abelian group. Thisresults are applied to the problem of approximating the inverse Radon Transform of a function.Keywords: LCA groups, random measures and integrals, Radon Transform.2000 AMS Subject Classification: 21A54 - 55P541 INTRODUCTIONLet G be an LCA group, i. e. a locally compact abelian group with Hausdorff topology, whose groupoperation is written additively. Let U be a topological Hausdorff space and B(U) the Borel σ-algebra of U .The symbol 1S stands for the indicator function of the set S. Denote by Γ the dual group of G and by �γ, x�the value of γ ∈ Γ at x ∈ G. Let mG denote the Haar measure on G, i.e. the unique invariant measurewith respect to the group operation ‘+’. The respective Lebesgue spaces of functions, for p ∈ [1,∞], aredenoted by Lp(G). If f ∈ L1(G), its Fourier Transform at γ ∈ Γ is defined byFf(γ) = f̂(γ) =�Gf(x)�γ, x�dmG(x) .In a similar manner one can define a Haar measure mΓ over (Γ,B(Γ )), and then the inverse Fourier trans-form for g ∈ L1(Γ ) at x ∈ G is given by: F−1g(x) = g∨(x) =�Γg(γ)�γ, x�dmΓ (γ) . A classical exampleis G = Rd with its usual addition operation +, and its dual Γ = Rd. In this case �γ, x� = eiγ.x withmG and mΓ the ordinary Lebesgue measure on Rd. Another very known example is the torus G = T andits dual Γ = Z. In this case, �γ, x� = eiγ.x with mG again the Lebesgue measure and mΓ the countingmeasure. With these basic definitions in hand, the most relevant results for these classical cases, such asPlancherel’s formula, can be extended with no difficulty to the general abstract setting of an LCA groupG [4]. A key result in Harmonic Analysis and Signal Processing for G = Rd is the so called Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem which gives conditions to reconstruct (interpolate) a bandlimited L2(Rd) function from its discrete samples taken at a uniform and appropriate rate. The WKS theo-rem was extended to L2(G), with G an LCA group, by Kluvánek [5]. In addtion to its elegance, Kluvanek’sresult provides an example of a unified theory which gives a positive answer to several similar problems thatmay seem different at first glance. The WKS theorem and its abstract version by Kluvanek, are examplesof uniform sampling theorems. In practice, this means that if f represents a signal its samples are takenat equidistant points, in the abstract setting this traduces in evaluating f over a discrete subgroup H ⊂ G.However, in some applications, non uniform sampling may be preferable to the uniform sampling approach.An interesting case arises when the sample points are chosen randomly. Several authors studied the case ofsampling points given by a Poisson point process over Rd, see e.g.[8]. Poisson point processes or relatedRandom measures can be defined over general σ-finite measure spaces [6]. In particular one can considerthe measure space (G,B(G),mG). Here, we shall study some results for this abstract sampling setting andthen we will apply these to the particular case of G = R×T arising in the context of the Radon transform.MACI Vol. 8 (2021) M.L. Schuverdt, N.L. Kudraszow, R.P. Vignau, M.D. Sánchez (Eds.)2572 SOME PRELIMINARIESWe begin by recalling some definitions. Let G be a LCA group. Analogously to the Rd case, for ameasurable S ⊂ Γ, one can define the Paley-Wiener spaces of S-band limited functions as: PWS = {f ∈L2(G) : supp(f̂) ⊂ S} . If mΓ (S) < ∞, we call kS := F−11S . Then the orthogonal projection over PWSis the operator TS : L2(G) −→ L2(G) defined asTSf(x) = kS ∗ f(x) =�Gf(t)kS(x− t)dmG(t) =�Sf̂(γ)�γ, x�dmΓ (γ) .Let (Ω,A,P) be a probability space and X a random variable defined on it. If ϕ is any Borel measurablereal function, we denote E(ϕ(X)) the expectation of ϕ(X). If B0(G) = {A ∈ B(G) : mG(A) < ∞}and λ > 0, we can define a Poisson random measure [6] N : B0(G) −→ L2(Ω,G,P), where G is thesub σ-algebra of A generated by the familty of random variables {N(A), A ∈ B0(G)} and such thatE(N(A)) = λmG(A) and V ar(N(A)) = λmG(A). We aim to define an stochastic integral for somesuitable random integrands. Let F be a sub σ-algebra of A and independent of G. If p > 0, we define thespacesLp(F) = {f : Ω×G −→ C is F ⊗B(G)- measurable and �f�Lp < ∞} ,where �f�pLp = E �f�pLp(G). Now, let R = {R = F ×B, F ∈ F , B ∈ B0(G)} be the class of measurablerectangles. From N we define a new random measure M : R −→ L2(Ω,σ(F ∪ G),P), byM(R) = M(F ×B) = N(B)1F . (1)If f is a R-simple function, f = �Ni=i ai1Ri with ai ∈ C and the Ri ∈ R disjoint, we define the stochasticintegral of f with respect to M as the random variable:�Gf(x)dM(x) :=N�i=iaiM(Ri) .The integral�G f(x)dM(x) verifies the following properties:Lemma 1 Let f be a R- simple function, then:i) E��G f(x)dM(x)�= λ�GE(f(x))dmG(x).ii) E��G f(x)dM(x)− λ�GE(f(x))dmG(x)�2= λ�GE|f(x)|2dmG(x) .By a limiting process, (ii) allows to extend the definition of�G f(x)dM(x) for any f ∈ L1(F)�L2(F).Moreover, Lemma 1 remains true for f ∈ L1(F)�L2(F).3 RANDOM SAMPLING ON GLet {Gn}n∈N0 be a sequence of independent sub σ-algebras of A, and let {Fn}n∈N be the sequence ofsub σ-algebras defined as Fn = σ(∪nj=0Gj). Let {Nn}n∈N be a sequence of Poisson random measures ofparameter λ > 0 such that Nn is Gn-measurable, and let Mn be defined as in equation (1), for each n. IfmΓ (S) < ∞ , then for f ∈ L2(Fn−1), we define:TnSf(x) =1λ�Gf(t)kS(x− t)dMn(t), (2)for each n ∈ N. We summarize the main facts about TnS in the following results:Theorem 1 Let f, g ∈ L2(Fn−1) then:i) TnSf ∈ L2(Fn) and TnSf ∈ PWS a.s..ii) If γ > 0 then:E �f − γTnSg�2L2(G) = E �f − γTSg�2 +γ2mΓ (S)λE �g�2L2(G) .MACI Vol. 8 (2021) M.L. Schuverdt, N.L. Kudraszow, R.P. Vignau, M.D. Sánchez (Eds.)258Note that for fixed n, if f = g ∈ PWS a.s. and γ = 1 thenE �f − TnSf�2L2(G) =mΓ (S)λE �f�2L2(G) .In this case, note that if λ −→ ∞ then �f − TnSf�L2(G) → 0, in the mean square sense and in probability.Therefore TnSf is an unbiased estimator of f . However, to ensure a good approximation of f the averagenumber of samples determined by λ should be considerably large. Theorem 1 and its consequent derivationslead to an improvement by considering an iterative method. For determinstic f ∈ L2(G), define a sequenceof functions {fn}n∈N0 , fn ∈ L2(Fn) ∩ PWS by: If n = 0, take f0 = 0 and if n ≥ 1:fn+1 = fn + γTn+1S(f − fn) . (3)The main result then reads:Theorem 2 Let f ∈ PWS (deterministic) and fn be defined by equation (3) then:E �f − fn�2L2(G) ≤ δ(γ)n �f�2L2(G) , (4)where 0 < δ(γ) =�(1− γ)2 + γ2mΓ (S)λ�. In particular, for fixed λ > 0, γ can be chosen such thatδ(γ) < 1 and thus equation (4) tends to 0 exponentially when n −→ ∞.4 RADON TRANSFORM AND THE GROUP G = R× TAssume that f ∈ C∞0 (R2) with compact support. Its Radon Transform Rf : R × [0, 2π) −→ R isdefined by:Rf(p, θ) =�{x: ξ·x=p}f(x) dx =�R2δ(p− ξ · x)f(x) dx, ξ(θ) = (cos θ, sin θ), p ∈ R.Briefly, the Radon transform is defined by the integrals of f over the lines {x : ξ · x = p}. Obviously,R is a linear transformation and, for fixed p, Rf(p, θ) can be viewed as a periodic function. At this point,following [2, 7], it is possible to apply the tools developed in the previuos section in case of the groupG = R × T. The projection-slice [1] theorem can be used to prove the following approximate inversionformula [2]:Theorem 3 Let e ∈ L2(R2) be a radial function such that |t|1/2ê(t) ∈ L2(R2) and ψ the even functionof one variable given ê(t) = (2π)−1ψ(|t|). Let the associated convolution kernel k be given by k̂(r) =12(2π)−3/2|r|ψ(r). Then:e ∗ f(x) =�R�[0,2π)k(ξ · x− p)Rf(p, θ) dθ dp.In fact, e ∗ f can be viewed as a filtered or smoothed version of f . If e defines an approximation of theidentity then the formula gives an approximation of f . Generally, this filtered version is easier to recoverfrom data Rf than the original function f . If F = R × [0, 2π) −→ R and if F1F denotes the Fouriertransform on the first variable, we define the integral operator Tψ, for x ∈ R2 as:TψF (x) =�R�[0,2π)|r|ψ(r)F1F (r, θ) drdθ .Indeed, one can prove that:e ∗ f(x) = (TψRf)(x) (5)Recall that now G = R× T. The key property of Tψ is the following:MACI Vol. 8 (2021) M.L. Schuverdt, N.L. Kudraszow, R.P. Vignau, M.D. Sánchez (Eds.)259Lemma 2 Let F ∈ L2(G), then |TψF (x)| ≤ (2π)1/2 �|r|ψ(r)�L2(R) �F�L2(G) .Now, we can combine equation (5) and Lemma 2 with Theorem 2 in the following way. Suppose thatF = Rf ∈ PWS , where S ⊂ Γ = R × Z is such that mΓ (S) < ∞. Note that, in this case, mΓ is theproduct measure between the Lebesgue measure mR and the counting measure c, mΓ = mR ⊗ c. Finallydefine a sequence {Fn}n as in equation (3), then we can prove:Theorem 4 Let F = Rf ∈ PWS and Fn be defined by equation (3) then:E �e ∗ f − TψFn�2L∞(R2) ≤ δ(γ)n(2π)1/2 �|r|ψ(r)�L2(R) �F�L2(G) , (6)where 0 < δ(γ) is defined as in Theorem 2.To illustrate the proposed method we include some experimental results. On the top we can see asmoothed version of CT image of a brain and the sinogrogram of the original image. Below, on the rightside we can see the reconstruction of the original Radon transform after 7 iterations, taking λ = 0.04 andγ = 0.9. On the left, the final approximation of the smoothed image.REFERENCES[1] S.R. DEANS, The Radon Transform and Some of Its Applications, Dover, 1993.[2] A. FARIDANI AND E.L. RITMAN, High-Resolution computed tomography from efficient sampling, Inverse Problems, 16,pp.635-650, 2000.[3] E. HEWITT AND K. A. ROSS, Abstract Harmonic Analysis. Volume II: Structure and analysis for compact groups. Analysison locally compact Abelian groups, Die Grundlehren der Mathematischen Wissenschaften, Band 152, Springer-Verlag, Berlin-New York, 1970.[4] E. HEWITT AND K. A. ROSS, Abstract Harmonic Analysis. Volume I: Structure of topological groups, integration theory,group representations, Second edition. Die Grundlehren der Mathematischen Wissenschaften. Band 115. Springer-Verlag,Berlin-New York, 1979.[5] I. KLUVÁNEK, Sampling theorem in abstract harmonic analysis, Matematicko-Fyzikalny Casopis, 15, 1, pp. 43-47, 1965.[6] T.G. KURTZ, Lectures on Stochastic Analysis, University of Wisconsin, 2007.(online:http://www.math.wisc.edu/ kurtz/735/main735.pdf)[7] F. NATTERER, Sampling in Fan Beam Tomography, SIAM J. Appl. Math. Vol. 53, 2, pp. 358-380, 1993.[8] A. PAPOULIS AND U. PILLAI, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 2002.MACI Vol. 8 (2021) M.L. Schuverdt, N.L. Kudraszow, R.P. Vignau, M.D. Sánchez (Eds.)260

Random sampling over lca groups and inversion of the radon transform

https://ri.conicet.gov.ar/bitstream/11336/156239/2/CONICET_Digital_Nro.5628470c-869c-4f6b-b820-89d841d26b0d_A.pdf

Random sampling over lca groups and inversion of the radon transform

Abstract

Similar works

Full text

Available Versions

CONICET Digital